This paper develops an asymptotic theory for a general class of nonlinear nonstationary regressions, extending earlier work by Phillips and Hansen (1990) on linear cointegrating regressions. The model considered accommodates a linear time trend and stationary regressors, as well as multiple I(1) regressors. We establish consistency and derive the limit distribution of the nonlinear least squares estimator. The estimator is consistent under fairly general conditions but the convergence rate and the limiting distribution are critically dependent upon the type of the regression function. For integrable regression functions, the parameter estimates converge at a reduced n 1 /4 rate and have mixed normal limit distributions. On the other hand, if the regression functions are homogeneous at inﬁnity, the convergence rates are determined by the degree of the asymptotic homogeneity and the limit distributions are non-Gaussian. It is shown that nonlinear least squares generally yields ineﬀicient estimators and invalid tests, just as in linear nonstationary regressions. The paper proposes a methodology to overcome such diﬀiculties. The approach is simple to implement, produces eﬀicient estimates and leads to tests that are asymptotically chi-square. It is implemented in empirical applications in much the same way as the fully modiﬁed estimator of Phillips and Hansen.
Chang, Yoosoon; Park, Joon Y.; and Phillips, Peter C.B., "Nonlinear Econometric Models with Cointegrated and Deterministically Trending Regressors" (1999). Cowles Foundation Discussion Papers. 1495.